| Title:
|
A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations (English) |
| Author:
|
Naito, Manabu |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
3 |
| Year:
|
2024 |
| Pages:
|
317-336 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The half-linear differential equation $$ (|u'|^{\alpha }{\rm sgn} u')' = \alpha (\lambda ^{\alpha + 1} + b(t))|u|^{\alpha }{\rm sgn} u, \quad t \geq t_{0}, $$ is considered, where $\alpha $ and $\lambda $ are positive constants and $b(t)$ is a real-valued continuous function on $[t_{0},\infty )$. It is proved that, under a mild integral smallness condition of $b(t)$ which is weaker than the absolutely integrable condition of $b(t)$, the above equation has a nonoscillatory solution $u_{0}(t)$ such that $u_{0}(t) \sim {\rm e}^{- \lambda t}$ and $u_{0}'(t) \sim - \lambda {\rm e}^{- \lambda t}$ ($t \to \infty $), and a nonoscillatory solution $u_{1}(t)$ such that $u_{1}(t) \sim {\rm e}^{\lambda t}$ and $u_{1}'(t) \sim \lambda {\rm e}^{\lambda t}$ ($t \to \infty $). (English) |
| Keyword:
|
half-linear differential equation |
| Keyword:
|
nonoscillatory solution |
| Keyword:
|
asymptotic form |
| MSC:
|
34C11 |
| MSC:
|
34D05 |
| MSC:
|
34D10 |
| idZBL:
|
Zbl 07953706 |
| idMR:
|
MR4801105 |
| DOI:
|
10.21136/MB.2023.0158-22 |
| . |
| Date available:
|
2024-09-11T13:45:36Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152537 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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